Integrand size = 17, antiderivative size = 74 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=-3 \sqrt [3]{x}+2 \sqrt {x}+\frac {6 x^{5/6}}{5}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [6]{x}}{\sqrt {3}}\right )-4 \log \left (1+\sqrt [6]{x}\right )-\log \left (1-\sqrt [6]{x}+\sqrt [3]{x}\right ) \] Output:
-3*x^(1/3)+2*x^(1/2)+6/5*x^(5/6)-2*3^(1/2)*arctan(1/3*(1-2*x^(1/6))*3^(1/2 ))-4*ln(1+x^(1/6))-ln(1-x^(1/6)+x^(1/3))
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=-3 \sqrt [3]{x}+2 \sqrt {x}+\frac {6 x^{5/6}}{5}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [6]{x}}{\sqrt {3}}\right )-4 \log \left (1+\sqrt [6]{x}\right )-\log \left (1-\sqrt [6]{x}+\sqrt [3]{x}\right ) \] Input:
Integrate[(1 + x^(1/3))/(1 + Sqrt[x]),x]
Output:
-3*x^(1/3) + 2*Sqrt[x] + (6*x^(5/6))/5 - 2*Sqrt[3]*ArcTan[(1 - 2*x^(1/6))/ Sqrt[3]] - 4*Log[1 + x^(1/6)] - Log[1 - x^(1/6) + x^(1/3)]
Time = 0.57 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {7267, 2027, 2426, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt [3]{x}+1}{\sqrt {x}+1} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 6 \int \frac {x^{7/6}+x^{5/6}}{\sqrt {x}+1}d\sqrt [6]{x}\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle 6 \int \frac {\left (\sqrt [3]{x}+1\right ) x^{5/6}}{\sqrt {x}+1}d\sqrt [6]{x}\) |
\(\Big \downarrow \) 2426 |
\(\displaystyle 6 \int \left (\frac {\sqrt [6]{x} \left (1-\sqrt [6]{x}\right )}{\sqrt {x}+1}+x^{2/3}+\sqrt [3]{x}-\sqrt [6]{x}\right )d\sqrt [6]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 \left (-\frac {\arctan \left (\frac {1-2 \sqrt [6]{x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {x^{5/6}}{5}+\frac {\sqrt {x}}{3}-\frac {\sqrt [3]{x}}{2}-\frac {2}{3} \log \left (\sqrt [6]{x}+1\right )-\frac {1}{6} \log \left (\sqrt [3]{x}-\sqrt [6]{x}+1\right )\right )\) |
Input:
Int[(1 + x^(1/3))/(1 + Sqrt[x]),x]
Output:
6*(-1/2*x^(1/3) + Sqrt[x]/3 + x^(5/6)/5 - ArcTan[(1 - 2*x^(1/6))/Sqrt[3]]/ Sqrt[3] - (2*Log[1 + x^(1/6)])/3 - Log[1 - x^(1/6) + x^(1/3)]/6)
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {6 x^{\frac {5}{6}}}{5}+2 \sqrt {x}-3 x^{\frac {1}{3}}-\ln \left (1-x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )+2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{6}}-1\right ) \sqrt {3}}{3}\right )-4 \ln \left (1+x^{\frac {1}{6}}\right )\) | \(56\) |
default | \(\frac {6 x^{\frac {5}{6}}}{5}+2 \sqrt {x}-3 x^{\frac {1}{3}}-\ln \left (1-x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )+2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{6}}-1\right ) \sqrt {3}}{3}\right )-4 \ln \left (1+x^{\frac {1}{6}}\right )\) | \(56\) |
meijerg | \(2 \sqrt {x}-2 \ln \left (1+\sqrt {x}\right )-\frac {3 x^{\frac {1}{3}} \left (-8 \sqrt {x}+20\right )}{20}+2 x^{\frac {1}{3}} \left (-\frac {\ln \left (1+x^{\frac {1}{6}}\right )}{x^{\frac {1}{3}}}+\frac {\ln \left (1-x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )}{2 x^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{6}}}{2-x^{\frac {1}{6}}}\right )}{x^{\frac {1}{3}}}\right )\) | \(85\) |
Input:
int((1+x^(1/3))/(1+x^(1/2)),x,method=_RETURNVERBOSE)
Output:
6/5*x^(5/6)+2*x^(1/2)-3*x^(1/3)-ln(1-x^(1/6)+x^(1/3))+2*3^(1/2)*arctan(1/3 *(2*x^(1/6)-1)*3^(1/2))-4*ln(1+x^(1/6))
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} + 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} - \log \left (x^{\frac {1}{3}} - x^{\frac {1}{6}} + 1\right ) - 4 \, \log \left (x^{\frac {1}{6}} + 1\right ) \] Input:
integrate((1+x^(1/3))/(1+x^(1/2)),x, algorithm="fricas")
Output:
2*sqrt(3)*arctan(2/3*sqrt(3)*x^(1/6) - 1/3*sqrt(3)) + 6/5*x^(5/6) + 2*sqrt (x) - 3*x^(1/3) - log(x^(1/3) - x^(1/6) + 1) - 4*log(x^(1/6) + 1)
Result contains complex when optimal does not.
Time = 2.75 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.09 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=\frac {16 x^{\frac {5}{6}} \Gamma \left (\frac {8}{3}\right )}{5 \Gamma \left (\frac {11}{3}\right )} - \frac {8 \sqrt [3]{x} \Gamma \left (\frac {8}{3}\right )}{\Gamma \left (\frac {11}{3}\right )} + 2 \sqrt {x} - 2 \log {\left (\sqrt {x} + 1 \right )} - \frac {16 e^{- \frac {2 i \pi }{3}} \log {\left (- \sqrt [6]{x} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {8}{3}\right )}{3 \Gamma \left (\frac {11}{3}\right )} - \frac {16 \log {\left (- \sqrt [6]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac {8}{3}\right )}{3 \Gamma \left (\frac {11}{3}\right )} - \frac {16 e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [6]{x} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {8}{3}\right )}{3 \Gamma \left (\frac {11}{3}\right )} \] Input:
integrate((1+x**(1/3))/(1+x**(1/2)),x)
Output:
16*x**(5/6)*gamma(8/3)/(5*gamma(11/3)) - 8*x**(1/3)*gamma(8/3)/gamma(11/3) + 2*sqrt(x) - 2*log(sqrt(x) + 1) - 16*exp(-2*I*pi/3)*log(-x**(1/6)*exp_po lar(I*pi/3) + 1)*gamma(8/3)/(3*gamma(11/3)) - 16*log(-x**(1/6)*exp_polar(I *pi) + 1)*gamma(8/3)/(3*gamma(11/3)) - 16*exp(2*I*pi/3)*log(-x**(1/6)*exp_ polar(5*I*pi/3) + 1)*gamma(8/3)/(3*gamma(11/3))
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{6}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} + 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} - \log \left (x^{\frac {1}{3}} - x^{\frac {1}{6}} + 1\right ) - 4 \, \log \left (x^{\frac {1}{6}} + 1\right ) \] Input:
integrate((1+x^(1/3))/(1+x^(1/2)),x, algorithm="maxima")
Output:
2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/6) - 1)) + 6/5*x^(5/6) + 2*sqrt(x) - 3*x^(1/3) - log(x^(1/3) - x^(1/6) + 1) - 4*log(x^(1/6) + 1)
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{6}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} + 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} - \log \left (x^{\frac {1}{3}} - x^{\frac {1}{6}} + 1\right ) - 4 \, \log \left (x^{\frac {1}{6}} + 1\right ) \] Input:
integrate((1+x^(1/3))/(1+x^(1/2)),x, algorithm="giac")
Output:
2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/6) - 1)) + 6/5*x^(5/6) + 2*sqrt(x) - 3*x^(1/3) - log(x^(1/3) - x^(1/6) + 1) - 4*log(x^(1/6) + 1)
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.28 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2\,\sqrt {x}+\ln \left (\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (27+\sqrt {3}\,9{}\mathrm {i}\right )+36\,x^{1/6}+36\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )-\ln \left (\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (-27+\sqrt {3}\,9{}\mathrm {i}\right )+36\,x^{1/6}+36\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )-4\,\ln \left (36\,x^{1/6}+36\right )-3\,x^{1/3}+\frac {6\,x^{5/6}}{5} \] Input:
int((x^(1/3) + 1)/(x^(1/2) + 1),x)
Output:
log((3^(1/2)*1i - 1)*(3^(1/2)*9i + 27) + 36*x^(1/6) + 36)*(3^(1/2)*1i - 1) - 4*log(36*x^(1/6) + 36) - log((3^(1/2)*1i + 1)*(3^(1/2)*9i - 27) + 36*x^ (1/6) + 36)*(3^(1/2)*1i + 1) + 2*x^(1/2) - 3*x^(1/3) + (6*x^(5/6))/5
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {1+\sqrt [3]{x}}{1+\sqrt {x}} \, dx=2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x^{\frac {1}{6}}-1}{\sqrt {3}}\right )+\frac {6 x^{\frac {5}{6}}}{5}-3 x^{\frac {1}{3}}+2 \sqrt {x}-2 \,\mathrm {log}\left (x^{\frac {1}{6}}+1\right )+\mathrm {log}\left (-x^{\frac {1}{6}}+x^{\frac {1}{3}}+1\right )-2 \,\mathrm {log}\left (\sqrt {x}+1\right ) \] Input:
int((1+x^(1/3))/(1+x^(1/2)),x)
Output:
(10*sqrt(3)*atan((2*x**(1/6) - 1)/sqrt(3)) + 6*x**(5/6) - 15*x**(1/3) + 10 *sqrt(x) - 10*log(x**(1/6) + 1) + 5*log( - x**(1/6) + x**(1/3) + 1) - 10*l og(sqrt(x) + 1))/5